Primitive elements in integral bases

On the basis of numerical observations H. Cohen and H. W. Lenstra, Jr. have posed the following question: does a Z-basis of a ring of integers in a number field necessarily contain a field generator of the number field? In this note it is shown that the answer is yes for all normal fields of prime power degree and for all fields whose degree is less than 12. For dihedral fields of degree 12 the answer is no. More generally, we consider the index in the ring of integers of the additive subgroup generated by integers from subfields. This index depends on subtle ramification phenomena, but one can give explicit formulas in certain cases by applying Fröhlich’s theory of factor equivalence.